ECOM032 ECONOMETRICS B 19/20
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Main Examination Period 19/20
ECOM032 ECONOMETRICS B
Consider n independent and identically distributed observations Xi. It is assumed that one of the two following hypotheses is true: under H0, the common distribution of the Xi is a normal distribution with unknown mean µ0 and variance σ0(2), denoted N╱µ0 , σ0(2)、, while under H1 the observations are not normal but have ﬁnite moments up to order 6. In what follows, θ = ╱µ, σ2、| with θ0 = ╱µ0 , σ0(2)、| , and
Consider a sequence Ω of symmetric nonnegative 3 x3 matrices, (θ) = (1 (θ) , 2 (θ) , 3 (θ))| and
GMM = arg min (θ)| (θ)
a) Suppose that H0 is true.
i) Show that the maximum likelihood estimator ML = ╱, 2、with
= ) Xi , 2 = ) (Xi - )2 ,
is the unique solution to the two equations 1 (θ) = 0, 2 (θ) = 0.
ii) Show that GMM = ML when
= ↓ 0(1) 1(0) ! 0 0
0 ! .
b) Suppose that H0 is true.
i) Show that, under H0 , 匝 [3 (θ0 )] = 0.
ii) Show that ,n (θ0 ) converges in distribution to a normal variable with a mean and variance to be given. Brieﬂy recall why it implies that ,n ╱GMM - θ0 ← converges in distribution to a normal with mean 0 and variance V (Ω) (do not attempt to give a closed form expression for V (Ω)).
iii) Compare the matrix V (Ω) with the inverse of the matrix I (θ0 )
deﬁned as follows
I (θ0 ) = ┌
iv) Describe two choices of GMM weight matrix 丰 Ω 丰 ensuring that V (Ω 丰 ) = I (θ0 )|1 (do not attempt giving explict expressions of the proposed 丰 ).
c) What are the relative merits of ML and GMM under H0?
d) Give the test statistic, its null limit distribution and the rejection region of the overidentiﬁcation test of H0 against H1 based on the moments (θ).
e) Is the test in part (d) consistent against all the alternatives in H1?
You are interested in the impact of the time spent on education Zi on the log wage wit of individual i = 1, . . . , N over time t = 1, . . . , T. You observe also a 1 x 1 control variable Xit which varies across i and t. It is assumed that
wit = γ0Zi(|) + β0Xit + ui + εit , 匝 [εit] = 0, Var (εit) = σε(2) ,
Cov (εi, Xit) = Cov (εi, Zi) = 0,
for some centered random variable ui possibly correlated with Zi but not with Xit .
a) Why, in this setup, ui be correlated with Zi?
b) Show that γ0 and β0 are a solution to the system
匝 [(wit - γZi - βXit) Xit] = 0,
匝 ┌(wit - γZi - βXit) Xi┐ = 0,
where Xi = ( Xit .
c) Let (γ, β) = [1 (γ, β) , 2 (γ, β)]| where
1 (γ, β) =
2 (γ, β) =
) ) (wit - γZi - βXit) Xit ,
) ) (wit - γZi - βXit) Xi .
Deﬁne the Generalised Method of Moments (GMM) estimator of γ0 and β0 based on the two moment conditions in Question b. Is it necessary to weight the moment conditions with a general weighting matrix Ω?
d) Assume the Xit’s are iid and independent of the error terms. Compute the variance matrix of ,N(γ0 , β0 ). Why is this variance impor- tant?
e) Compute the matrix
Does this matrix depend upon γ and β?
G = - – 匝(匝) (Xi(X）￥T)匝2(匝)『 .
Show that converges to G in probability when N diverges and T is ﬁxed, under some conditions to be stated. Show that G has an inverse if T 2 2, Var (Xit) 0 and 匝 [ZiXit] 0. Interpret the condition 匝 [ZiXit] 0.
g) Suppose T 2 2, Var (Xit) 0 and 匝 [ZiXit] 0. What is the asymptotic distribution of the GMM estimator introduced in Question c when N diverges and T is ﬁxed? Do not attempt to establish this limit distribution but use a result discussed in class.